Tagged Questions

Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Consider using one of the following tags: (model-theory), (set-theory), (computability), (proof-theory) if they fit the question.

I am a student and I get confused in translating some sentence to logic assertion.
For example: Joe does not have a lawyer, i.e. is not a customer of any lawyer.
The right way to translate is:
"For ...

The substitution formula sub( x,x,y) says
y is the code of the formula obtained when in the formula whose code is x, the numeral for x is substituted for the free variable.
Both x and y are free in ...

I interpreted this as a case of the extension of De Morgan's Law to quantifiers.
https://en.wikipedia.org/wiki/De_Morgan%27s_laws#Extensions
I know that similar questions have been asked before about ...

This is Velleman's exercise 3.3.10. Suppose that $\mathcal F$ is a nonempty family of sets, B is a set, and $\forall A \in \mathcal F (B\subseteq A)$. Prove that $B \subseteq \bigcap \mathcal F $.
My ...

Logic and model theory are not my area so my thinking is probably off, but I am curious about this so please go ahead and set me straight.
A definable set is one for which there is a formula that is ...

The use of the term "such that" confuses me I've seen this like $A=\{(x,y) :x,y\in\Bbb R\ \text{and } P(x,y) \}$ and $B=\{(x,y)\in \Bbb R^2:P(x,y)\}$ for some predicate $P$.
Is there any difference ...

Let $\mathbb N = \{ 1,2,3,\ldots \}$, then by the well-known "Cantor"-Scheme we have $\mathbb N \times \mathbb N \cong \mathbb N$. But even nicer is that we can write this scheme $\varphi : \mathbb N ...

The question is from Exercise 13 of part 1.4 in Rosen's "Discrete Mathematics and Its Applications" (5th edition): "let $M(x,y)$ be "$x$ has sent y an e-mail message", where the universe of discourse ...

I have stumbled upon the following reasoning, but I'm not sure if it's correct. Here it goes:
Domain X
$\forall x :\phi(x)⟹\gamma(x)$
Let $E\subseteq X⟹[\forall x\in E :\phi(x)⟹\gamma(x)]$
Suppose I ...

there's something that have been bugging me.
If we have quantities A, C, E
And if we have quantities B, D, F
And if we take the equimultiples G, H, K from A, C, E
And if we take the equimultiples L, ...

We say a cardinal $\theta$ is sufficiently large for a forcing $Q$ if $\mathcal{P}(\mathcal{P}(Q)) \in H(\theta)$.
And a set $M$ is a suitable model for $Q$ if $Q \in M$ and $M \prec H(\theta)$, $M$ ...

By Solovay's theorem, assuming the existence of an inaccessible cardinal, the axiom of choice is necessary to prove the existence of nonmeasurable sets. In the past, I've thought that one consequence ...

Denote by $C(n)$ the plain Kolmogorov complexity of $n$ and the length of a binary encoding of $n$ by $l(n)$, why do we have
$$
C(n\mid l(n)) \ge C(n) - C(l(n))?
$$
If I have a shortest program $p$ ...

In a Hilbert-style axiomatization of first-order logic (FOL), there is a rule for variable substitution
but I don't see any rule for substituting predicate symbols. Consider a theorem like:
$\forall ...